%% Analysis of Fixed-Point Numerically Controlled Oscillator % Numerically Controlled Oscillators (NCO) are efficient means of % generating sinusoidal signals. They are useful when a continuous phase % sinusoidal signal with variable frequency is required. % % This demo uses the Signal Processing Toolbox to analyze the NCO of a % Digital Down-Converter (DDC) implemented in fixed-point. Using spectral % analysis we will measure the Spurious Free Dynamic Range (SFDR) of the % NCO as well as explore the effects of adding phase dither. The number of % dither bits affects hardware implementation choices. Adjusting the number % of dither bits in simulation allows you to see the trade-offs between % noise floor level, spurious effects, and number of dither bits before % implementing in hardware. The DDC in the demo models the Graychip 4016 % and was designed to meet the GSM specification. % % ** %% Introduction % Digital Down-Converters (DDC) are a key component of digital radios. The % DDC performs the frequency translation necessary to convert the high % input sample rates found in a digital radio, down to lower sample rates % (baseband) for further and easier processing. Adhering to the GSM % specifications, in this example, the DDC's input rate is at 69.333 MHz % and it is tasked with downconverting its input rate down to 270 KHz. % % The DDC consists of a Numerically Controlled Oscillator (NCO) and a mixer % to quadrature down convert the input signal to baseband. The baseband % signal is then low pass filtered by a Cascaded Integrator-Comb (CIC) % filter followed by two FIR decimating filters to achieve the desired low % sample-rate ready for further processing. The final stage often includes % a resampler which interpolates or decimates the signal to achieve the % desired sample rate depending on the application. Further filtering can % be achieved with the resampler. A block diagram of a typical DDC is % shown below. One thing to note is that Simulink handles complex signals, % so we don't have to treat the I and Q channels separately. % % <> %% % This demo focuses on the analysis of the NCO. For a demo on designing % the three-stage, multirate, fixed-point filter chain and HDL code % generation refer to the following demo %% The Numerically Controlled Oscillator (NCO) % The digital mixer section of the DDC includes a multiplier and a NCO, % which provide channel selection or tuning for the radio. It is basically % a sine and cosine generator, creating complex values for each sine/cosine % pair. A typical NCO consists of four components, the phase accumulator, % the phase adder, the dither generator, and sine/cosine lookup table. % % Here's a typical NCO circuit modeled in Simulink, similar to what you % would see in Graychip's data sheet. open_system('ddcnco'); %% % Based on the input frequency the NCO's phase accumulator produces values % which are used to address a sine/cosine table lookup. The phase adder is % used to specify a phase offset which provides the ability to phase % modulate the output of the phase accumulator. Phase dithering is provided % by the Dither Generator to reduce amplitude quantization noise and % therefore improve the Spurious Free Dynamic Range of the NCO. The % sine/cosine lookup table produces the actual complex sinusoidal signal % and the output is stored in the variable nco_nodither. % % In the Graychip the tuning frequency is specified as a normalized value % relative to the chip's clock rate. So, for a tuning frequency of F the % normalized frequency is F/Fck, where Fck is the chip's clock rate. The % phase offset is specified in radians ranging between 0 and 2pi. In this % demo the normalized tuning frequency is set to 5/24 while the phase % offset is set to 0. The tuning frequency is specified as a 32 bit word % and the phase offset is specified as a 16 bit word. % % Since the NCO is implemented using fixed-point arithmetic, it experiences % undesirable amplitude quantization effects. These are numerical % distortions due to finite wordlength effects. Basically, we have % sinusoids being quantized giving rise to cumulative, deterministic, % periodic errors in the time domain, which appear as line spectra (spurs) % in the frequency domain. The amount of attenuation from the peak of the % signal of interest to the highest spur is the Spurious Free Dynamic % Range. % % The Spurious Free Dynamic Range (SFDR) of the Graychip is 106dB, however % the GSM spec requires that the SFDR be > 110dB. There are a couple of % ways to improve the SFDR, we will explore the use of adding phase % dither to the NCO. % % The Graychip's NCO contains a phase dither generator which is basically a % random integer generator, and is used to improve the purity of the % oscillator output. Dithering causes the unintended periodicities of the % quantization noise (leading to "spikes" in spectra, poor SFDR) to be % "spread" across a broad spectrum, effectively reducing these undesired % spectral peaks. Conservation of energy applies, however, the spreading % effectively raises the overall noise floor. So it's good for SFDR, but % only up to a certain point. % % Let's run the NCO model and analyze its output in MATLAB's workspace. % This model is not using dithering. sim('ddcnco'); whos nco* %% % The plot below displays the real part of the first 128 samples of the % output of the NCO, which is stored in the variable called nco_nodither. plot(real(squeeze(nco_nodither(1:128)))); grid title('Real part of NCO Output - No Dithering') ylabel('Amplitude'); xlabel('Samples'); set(gcf,'color','white'); %% Spectral Analysis of NCO Output % When performing spectral estimation, it is very important to understand % your data in order to choose the appropriate spectral estimation % technique. Given that we have a large data set (over 20,000 samples), we % can rely on an FFT based classical method, such as periodogram, to % calculate the spectral content of the output of the NCO. % % The signal has some randomness, however it's primarily sinusoidal, so we % will measure its mean-square spectrum, as opposed to the power spectral % density which is more appropriate to measure the power of random signals. % For a demo on measuring power refer to Below we use the % msspectrum method to calculate and plot the Mean-square Spectrum of the % NCO signal. %% % Define spectral analysis algorithm. h = spectrum.periodogram %% % Calculate and plot the Mean-square Spectrum. Fs = 69.333e6; msspectrum(h,real(nco_nodither),'Fs',Fs) set(gcf,'color','white'); %% % As expected, the Mean-square Spectrum plot shows a peak at 14.44 MHz, % which is the NCO's tuning frequency 5/24*Fs = 14.444 MHz. % % Notice however that the noise floor which is about -82 dB is too high to % meet the GSM specifications, which requires -110 dB or more. We know we % can improve this by adding phase dither, but before we add dither let's % take a closer look at our analysis. %% Choosing the right Window % The periodogram spectral analysis technique uses a rectangular window, % which although it provides good frequency resolution (i.e., it has a % narrow mainlobe bandwidth), it has a high noise floor. Multiplying the % NCO signal which is sinusoidal, by a rectangular window is the same as % convolving the two signals in the frequency domain. The convolution of a % sinusoidal signal's frequency response, which is a delta, by a % rectangular window, whose frequency response is a sin(x)/x, will result % in a sin(x)/x response centered at the frequency of the delta. So, % there's a smearing of the delta function in the frequency domain. The % noise floor will be the addition of the two signals. Therefore, what % we're seeing is the noise floor of the rectangular window, which is much % higher than the highest signal spur. % % To verify that the noise floor of the window is preventing us from seeing % the signal spurs, let's look at the time and frequency response of % rectangular window. We can design such a window using the Window Design % Tool, WinTool, but here we will use the command line functionality. % % Define and view the frequency response of a rectangular window. N = length(nco_nodither); wrect = sigwin.rectwin(N) %% wvtool(wrect) %% % If we zoom in or use data markers we see that the maximum attenuation the % rectangular window achieves is about 84 dB, which is roughly the noise % floor we saw in the spectrum plot of the NCO output. % % Since in our analysis we're not trying to resolve two sinusoids, but % instead we're looking for spectral content below 100 dB, let's use % a Von Hann window which provides over 100 dB of attenuation. whann = sigwin.hann(N) %% % Let's view the Hann window both in time and frequency domain. wvtool(whann) %% % Indeed the frequency domain plot on the right shows that the Hann window % exhibits a much lower noise floor. So, the Von Hann window is better % suited for this particular problem. Here are the results of using the % Hann window to calculate the spectral estimate of the NCO output. h.WindowName = 'Hann'; msspectrum(h,real(nco_nodither),'Fs',Fs); set(gcf,'color','white'); %% % Using a Hann window to window our NCO signal produced a much lower noise % floor clearly displaying the spurious peaks. Now we can measure the % Spurious Free Dynamic Range (SFDR) and look into ways to decrease the % spurious peaks by using phase dithering. % % We can zoom-in using the axis command to measure the difference between % the peak of the carrier and the highest spurious peak. axis([0 35 -5 0]) %% % The peak is roughly -3.25dB. Now let's zoom-in to the highest spur. axis([0 35 -120 -100]) %% % The highest spur is -110 dB. The Spurious Free Dynamic Range then is the % difference between these two values 110dB - 3.25dB = 106.75 dB. %% Exploring the Effects of Dithering % In order to explore the effects of adding dither to the NCO, I have % encapsulated the NCO circuit shown above in a subsystem and duplicated % the NCO subsystem three times. Then a different amount of dither was % selected in each NCO. The NCO allows a range of 1 to 19 bits of dither % to be specified, however we'll just try a few values. Running this % model will produce three different NCO outputs based on the amount of % dither added. open_system('ddcncowithdither') %% % Running the simulation will produce three signals in the MATLAB workspace % that we can then analyze using the same spectral analysis algorithm % defined above. We can run the simulation from the model or from the % command line using the sim command. sim('ddcncowithdither') %% % After the simulation completes we're left with the signals which are the % output of the NCOs with varying amounts of dithering. whos nco* %% % As already seen above, using a Von Hann window we can see the spurious % peaks, and as expected, since we're modeling a Graychip, the highest spur % is at about -110 dB. The Spurious Free Dynamic Range then is about 107 % dB, but our GSM spec requires that it be at least 110 dB. This is where % dithering will help. %% % Let's start by adding 3 bits of dithering. msspectrum(h,real(nco_dither3),'Fs',Fs) % Plot Mean-square Spectrum set(gcf,'color','white'); %% % Again, let's zoom-in using the axis command to measure the difference % between the peak of the carrier and the highest spurious peak. axis([0 35 -5 0]) %% % As expected the peak is again roughly -3.25 dB. Now let's zoom-in to the % highest spur. axis([0 35 -120 -100]) %% % With three bits of dither added, the highest spur is now about 112 dB. % The SFDR then is 112dB - 3.25dB = 108.75 dB. Still doesn't meet the GSM % specification which requires an SFDR of 110 dB or more. % %% % Let's try adding 5 bits of dithering. msspectrum(h,real(nco_dither5),'Fs',Fs) % Plot Mean-square Spectrum set(gcf,'color','white'); %% % The peak of the carrier frequency should still be roughly -3.25 dB. Now % let's zoom-in to the highest spur. axis([0 35 -130 -120]) %% % With five bits of dither added, the highest spur is now about -127 dB. % The SFDR then is 127dB - 3.25dB = 123.75 dB. This SFDR meets and even % exceeds the GSM specification. % %% % So far the more dither we added the better the results were. So, let's % try adding 7 bits of dithering. msspectrum(h,real(nco_dither7),'Fs',Fs) % Plot Mean-square Spectrum set(gcf,'color','white'); %% % Zooming in on the noise floor we notice that highest spur is around % -122.5 dB, which results in an SFDR of 119.25 dB. This meets the GSM % spec, but we can do better using 5 bits of dithering. axis([0 35 -130 -120]) %% Comparing Results % Let's tabulate the SFDR for each NCO output given the various amounts of % dithering. %% % % Number of Spur Free Dynamic % Dither bits Range(dBc) % %% % % 0 107 % % 3 109 % % 5 123 % % 7 119 % %% Summary % In this demo we analyzed the output of a Numerically Controlled % Oscillator (NCO) used in a Digital Down Converter for a GSM application. % We used spectral analysis to measure the Spurious Free Dynamic Range % (SFDR), which is the difference between the highest spur and the peak of % the signal of interest. Spurs are deterministic, periodic errors that % result from quantization effects. We also explored the effects of adding % dither in the NCO, which is a process that adds random data to the NCO to % improve its purity. We found that using five bits of dithering achieved % the highest SFDR.